Problem: Find the smallest value of $x$ that satisfies the equation $|3x+7|=26$.
Answer: We can split the expression $|3x+7|=26$ into two separate cases: $3x+7=26$ and $3x+7=-26$. For the first case, solving for $x$ would give us $x=\frac{26-7}{3}=\frac{19}{3}$. For the second case, we would get $x=\frac{-26-7}{3}=\frac{-33}{3}=-11$. Therefore, $x=\frac{19}{3}$ and $x=-11$ both satisfy the equation. Since the problem asks for the smallest value of $x$, our solution is $\boxed{-11}$.